Upper join-closed subgroup property

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
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This article is about a general term. A list of important particular cases (instances) is available at Category:Upper join-closed subgroup properties

Definition

Definition with symbols

A subgroup property $p$ is said to be upper join-closed if given $H \le G$ and $K_i, i \in I$ are intermediate subgroups of $G$ containing $H$ (indexed by a nonempty set $I$) and $H$ satisfies $p$ in each $K_i$, we have that $H$ satisfies $p$ in the join of subgroups $\langle K_i \rangle_{i \in I}$.

Relation with other metaproperties

Stronger metaproperties

• Lower-intersection upper-join closed subgroup property
• LU-join closed subgroup property
• Upward-closed subgroup property
• Izable subgroup property

Weaker metaproperties

• Finite-upper join-closed subgroup property
• Permuting-upper join-closed subgroup property

Related notions

Given a subgroup property $p$ that is identity-true, upper join-closed and also satisfies the intermediate subgroup condition, we can, given any subgroup $H$ of $G$ associate a unique largest subgroup $M$ containing $H$ for which $H$ satisfies $p$ in $M$.

Such a subgroup property is termed an izable subgroup property and the $M$ that we get is termed the izing subgroup of $H$ for that subgroup property.

Properties satisfying it

Normality

Normality is an upper join-closed subgroup property, viz, if $H \le G$ and $K_1, K_2$ are intermediate subgroups such that $H \triangleleft K_1$ and $H \triangleleft K_2$, then $H \triangleleft <K_1,K_2>$.

Central factor

The property of being a central factor is also upper join-closed, in fact, it is izable.